Chapter 12 Fluid Flow

12-1 The Basic Equation of Steady-Flow The Conversation of Mass A ： Cross section area of flow duct c ： Velocity of fluid v ： Specific volume of fluid Differential form:

The Conversation of Energy Differential form:

Process Equation Usually the fluid flows too fast in a duct to exchange heat with its surroundings, so it undergoes an adiabatic process, then: pv k = const Differential form:

Velocity of Sound and Mach Number From Then The Velocity of Sound is denoted by

Mach Number Ma is defined as ：

Ernst Mach was born on Feb. 18, 1838, d. Vaterstetten. He found an experimental proof for the Doppler Effect (Christian Doppler) and by inspection of fast moving projects proposed the Mach's principle. The "Mach Number" named after him describes the relation of a body's velocity to sonic speed. His attitude was characterized by empirical thinking based on scientific findings. One of his strictest opponents and critics was M. Planck. After retiring from the University in 1901 he was appointed to the upper chamber of the Austrian Parliament, a post he held for 12 years. He died in 1916

12-2 The Fluid Flow in Duct During the fluid flow in a duct, the properties of fluid changes along the stream line. In most situations this can be treated as a one-dimensional flow The Pressure and Velocity From conservation of energy dh= - cdc du +d(pv)= - cdc du+pdv+vdp= -cdc

From the first law of thermodynamics: δq=du+pdv =0 Then: du+pdv+vdp= -cdc vdp= - cdc - vdp= cdc To increase the velocity of fluid( dc>0 ), the pressure must be decreased. This kind of duct is called Nozzle Or to decrease the velocity ( dc<0 ) of fluid to obtain a high pressure in a duct flow. This kind of duct is called diffuser

Velocity and The Cross Section Area of Duct From conservation of mass ： From the process equation

As to a nozzle dc >0 (1)If the fluid velocity is subsonic, then ( Ma 2 -1 )<0 Therefore: dA<0 The nozzle’s shape should be as following: Subsonic flow convergent nozzle

(2)If the fluid velocity is ultrasonic, then ( Ma 2 -1 )>0 Therefore: dA>0 The nozzle’s shape should be as following: Ultrasonic flow divergent nozzle

(3)If the nozzle’s inlet velocity is subsonic, but outlet velocity ultrasonic, then: dA 0 The nozzle’s shape should be as following: Subsonic flow Ultrasonic flow convergent-divergent nozzle throat

As to a diffuser dc <0 (1)If the fluid velocity is subsonic, then ( Ma 2 -1 )<0 Therefore: dA>0 The diffuser’s shape should be as following: Subsonic flow

(2)If the fluid velocity is ultrasonic, then ( Ma 2 -1 )>0 Therefore: dA<0 The diffuser’s shape should be as following: Ultrasonic flow

(2) If the diffuser’s inlet velocity is ultrasonic, but outlet velocity subsonic, then: dA 0 The diffuser’s shape should be as following: Ultrasonic flow Subsonic flow

Applications Ram-jet engine Diffuser(compressor) combustion chamber nozzle

Rocket Space Shuttle

12-3 The Calculation of Nozzle The Flux of Subsonic Nozzle We define P 2 /P 1 as compression ratio ε

The Critical Compression Ratio For convergent-divergent nozzle, the velocity at throat keeps as sound-velocity. This state is called critical state. The flux of this kind of nozzle is depend on that of throat. The Compression Ratio which is low enough to make the air flow at sound-velocity at the exit of nozzle is called the critical compression Ratio. It is denoted by ε c

εc=εc= for air for saturated steam for superheated steam

The End of This Book Thank You